The version of the notes I have is from 2006, they are organized in the form of a short book. It is my understanding they have been updated since, and I believe the current version has new material on model theory, computability, and incompleteness. In particular, I think that Woodin's proof of the second incompleteness theorem for set theory, that I have covered elsewhere, is discussed there.
I think that the notes are distributed to the students at Berkeley that take the course, usually taught by Ted or Hugh, but I do not know whether they plan to publish them, and I am not sure they want to disseminate them otherwise.
The table of contents of the version I have is as follows:
- Propositional logic
- First order logic: syntax
- First order logic: semantics
- The logic of first order structures
- Gödel's Completeness Theorem
- The Compactness Theorem
- More on the logic of structures
To give an idea of the content, the languages that are discussed are finite (or recursive), and set theoretical prerequisites are kept at a minimum. This simplifies the discussion of some key results (such as compactness or the Löwenheim-Skolem theorems). Besides what I have already mentioned, topics covered include elimination of quantifiers, model completeness, Presburger arithmetic, and a study of definability for particular structures.
I would expect that contacting Ted or Hugh directly is the best way to obtain a copy of the notes.